Bayesian Inference for Gaussian Process Classifiers with Annealing and Pseudo-Marginal MCMC

Kernel methods have revolutionized the fields of pattern recognition and machine learning. Their success, however, critically depends on the choice of kernel parameters. Using Gaussian process (GP) classification as a working example, this paper focuses on Bayesian inference of covariance (kernel) parameters using Markov chain Monte Carlo (MCMC) methods. The motivation is that, compared to standard optimization of kernel parameters, they have been systematically demonstrated to be superior in quantifying uncertainty in predictions. Recently, the Pseudo-Marginal MCMC approach has been proposed as a practical inference tool for GP models. In particular, it amounts in replacing the analytically intractable marginal likelihood by an unbiased estimate obtainable by approximate methods and importance sampling. After discussing the potential drawbacks in employing importance sampling, this paper proposes the application of annealed importance sampling. The results empirically demonstrate that compared to importance sampling, annealed importance sampling can reduce the variance of the estimate of the marginal likelihood exponentially in the number of data at a computational cost that scales only polynomially. The results on real data demonstrate that employing annealed importance sampling in the Pseudo-Marginal MCMC approach represents a step forward in the development of fully automated exact inference engines for GP models.